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I have an idea of what to do. I know that for $\gcd\left(a,\:10\right)\:=\:a$ to hold true, $a\le 10$ and $10$ also has to be a multiple of $a$.

Therefore, I think the only possible integers are $\pm 1,\:\pm 2,\pm 5,\:\pm 10$ but I'm not quite sure on how to prove this statement.

Any assistance would be appreciated!

someman112
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    "and $a$ also has to be a multiple of $10$" should be "and $a$ should be a divisor of $10$". – plop Nov 29 '21 at 21:54
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    What about $\pm 2$? It's just a question of listing the divisors of $10$. – lulu Nov 29 '21 at 21:55
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    According to the definition I'm used to, $\gcd(-5,10)=5$, not $-5$. – Greg Martin Nov 29 '21 at 21:59
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    Depends on how you define $\gcd,$ I suppose, but usually $\gcd$ is defined to be positive (or non-negative, if you define $\gcd(0,0)=0.$) – Thomas Andrews Nov 29 '21 at 21:59
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    @lulu forgot about 2, I edited the post. – someman112 Nov 29 '21 at 22:10
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    @GregMartin I included the negatives because the question asked for integers. I assumed that if it was only naturals then the questions would've been something like 'find all the naturals....' – someman112 Nov 29 '21 at 22:11
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    @Yaya123, by most conventions, the gcd is necessarily positive, since it's usually defined as the greatest integer that divides both arguments, hence greater than or equal to $1$ (which always divides both arguments). That would rule out negative values of $a$. – Barry Cipra Nov 29 '21 at 22:15
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    $\gcd(0,0)$ should definitely not equal $0$ (every positive integer is a common divisor of $0$ and $0$). – Greg Martin Nov 29 '21 at 22:37

1 Answers1

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To give this question slightly more mathematical content:

Proposition. For all $a, b \in \mathbb{N}$ we have $\operatorname{gcd}(a, b) = a$ if and only if $a \mid b$.

Proof. If $\operatorname{gcd}(a, b) = a$ then in particular $a \mid b$. On the other hand if $a \mid b$ then $\operatorname{gcd}(a, b) \geq a$, but $\operatorname{gcd}(a, b)$ cannot exceed $a$ so we have equality.

(So the answers are the positive factors of $b = 10$.)

Greg Martin
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Keeley Hoek
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